Unfortunately we don't have the Enterprise prop in front of us.
Your going to take your time opening it up and seeing if you can find a way to fit at least the front part of a 12mm copper module which is neccesary for that 9mm diode and like mentioned wire it to the driver somewhere else. The size of that prop doesn't look like it can fit a complete module with the driver attached to the back of the diode. I could be wrong??
From what iv'e seen from the TV show the phaser's beam's fire out just above the bulb part on the bottom. I would run the diode no more than 1.8A for a little less heat and still get close to 2W because there is no room for heatsinking the 12mm module. Still 5 second blasts like you want anyways.
Sorry to say but another option could be a 532nm, iv'e seen them before in a module smaller than 12mm. Can't remember but they are out there. I'll try to find some sources when I can.
532's are the most visible and 50mW of it is plenty bright. If you can try figure a way to fit 2 of these smaller 532 modules and angle them proper or just come to maby except just the one beam
Your original plan can be done though as iv'e seen plenty of creative talent from members and other prop makers..
Let's see if Lifetime17 has a 638nm twin beam he can setup or you just might have to order it from O-like and go from there..
Edit: Ebay has some 8mm and 6.5mm 532nm modules.. They are rated at anywhere from 5mw to 20mw but probably output more. Some of the pic's show the driver with a pot screw which "may" mean you can push them a little more. Just something to consider that all